I explained this at length with a blow by blow history. But yes, floor crossings:
In early 2005 the NDP pushed for significant NDP priorities. These went through only thanks to Belinda Stronach being induced to cross the floor and the dramatic vote of a dying Chuck Cadman for the budget..
The sponsorship scandal broke and worsened as the eyar went on: three Liberals crossed the floor with two of them saying they would vote against the Martin government specifically due to the scandal. (The third voted with Martin.)
At the time the NDP had an MP that lost the nomination and sat as an independent. She also vowed to vote against Martin and was unreachable to Layton.
Layton tried to negotiate a new arrangement for the 2006 budget with Martin -- focusing on health care. Martin told Layton to buzz off because he no longer had the votes.
Harper, the BQ and these independents had declared that they were going to defeat the government. Harper was attacking the NDP daily for propping up Harper saying his party should be defeated as well. Layton tried to negotiate for time to delay the fall of the government but Martin would have nothing to do with that.
The idea that Harper needed the NDP was false -- the BQ and even Liberal defectors were committed to bring down the government as well as Harper. In fact Harper would ahve preferred the NDP not support the motion as he clearly intended to use the NDP support of Martin fro earlier in the year in the campaign -- and he said so. Harper did not need a seconder from another party but he likely would have used one of the Liberal defectors (who left the Liberal caucus due to the scandal and vowed to vote down the government) if Layton had not cooperated.
Layton, seeing there was nothing left to do moved to distance himself and supported the non-confidence motions.
The non confidence motion was passed 171-133. The NDP had 18 members. You can see if they had voted with Martin the Liberals would have been defeated 153-151. The Liberals had 133 members but one was the Speaker -- however, one independent Liberal voted with Martin. there were two vacancies including Cadman who had voted with the government before but had later died.
It is an absolute lie to suggest that Layton and the NDP could have prevented the defeat of Martin. There was blood in the water and both the BQ and the Conservatives wanted the defeat and were not going to stop given they had been trying to defeat the government for a year. Only with the death of Cadman and two Liberal and one NDP defection to the side of defeating the government did that become possible. As I say, when Layton saw it was inevitable and heard the rhetoric from Harper about the NDP protecting Liberal sleaze, he put a knife into the already dead body of the Liberal government. It is false to say the NDP took the Liberals down. It is accurate to say they were brought down by their own defections due to the sponsorship scandal.
After the defeat, there were reports that the Liberals planned to use this defeat against the NDP by blaming them for it in spite of the reality of the numbers. Layton's enthusiasm -- designed to create distance was to be used against him. Liberals like Pondering have flogged this talking point ever since. However, the Liberal war rooms stopped when people rightly pointed out that it was bullshit. However, some disaffected New Democrats and journalists either through sloppiness or design neglect to remember all these facts and the numbers and repeat it to this day.
This should be a sticky -- thanks to Pondering we keep coming back to this even though the facts are completely against her.
I have in the past named each of the independents and specific dates but this is the summary.